# Ice cream and orbifold Riemann-Roch

• Published in 2012
In the collection
We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold X,D, under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called "ice cream functions". This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [A. Buckley and B. Szendroi, Orbifold Riemann-Roch for 3-folds with an application to Calabi-Yau geometry, J. Algebraic Geometry 14 (2005) 601--622] and [Shengtian Zhou, Orbifold Riemann-Roch and Hilbert series, University of Warwick PhD thesis, March 2011, 91+vii pp.], although the correct statements are considerably trickier. We expect to return to this in future publications.

## Other information

key
IcecreamandorbifoldRiemannRoch
type
article
2020-10-16
date_published
2012-11-14

### BibTeX entry

@article{IcecreamandorbifoldRiemannRoch,
key = {IcecreamandorbifoldRiemannRoch},
type = {article},
title = {Ice cream and orbifold Riemann-Roch},
author = {Anita Buckley and Miles Reid and Shengtian Zhou},
abstract = {We give an orbifold Riemann-Roch formula in closed form for the Hilbert
series of a quasismooth polarized n-fold X,D, under the assumption that X is
projectively Gorenstein with only isolated orbifold points. Our formula is a
sum of parts each of which is integral and Gorenstein symmetric of the same
canonical weight; the orbifold parts are called "ice cream functions". This
form of the Hilbert series is particularly useful for computer algebra, and we
illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds.
These results apply also with higher dimensional orbifold strata (see [A.
Buckley and B. Szendroi, Orbifold Riemann-Roch for 3-folds with an application
to Calabi-Yau geometry, J. Algebraic Geometry 14 (2005) 601--622] and
[Shengtian Zhou, Orbifold Riemann-Roch and Hilbert series, University of
Warwick PhD thesis, March 2011, 91+vii pp.], although the correct statements
are considerably trickier. We expect to return to this in future publications.},
comment = {},
}